Compact Two-point Homogeneous Spaces Research Articles

Single radius spherical cap discrepancy on compact two-point homogeneous spaces

In this note we study estimates from below of the single radius spherical discrepancy in the setting of compact two-point homogeneous spaces. Namely, given a d-dimensional manifold M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {M}$$\\end{document} endowed with a distance ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document} so that (M,ρ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mathcal {M}, \\rho )$$\\end{document} is a two-point homogeneous space and with the Riemannian measure μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}, we provide conditions on r such that if Dr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_r$$\\end{document} denotes the discrepancy of the ball of radius r, then, for an absolute constant C>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C>0$$\\end{document} and for every set of points {xj}j=1N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{x_j\\}_{j=1}^N$$\\end{document}, one has ∫M|Dr(x)|2dμ(x)⩾CN-1-1d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\int _{\\mathcal {M}} |D_{r}(x)|^2\\, d\\mu (x)\\geqslant C N^{-1-\\frac{1}{d}}$$\\end{document}. The conditions on r that we have depend on the dimension d of the manifold and cannot be achieved when d≡1(mod4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\equiv 1 \\ ( {\ ext {mod}}4)$$\\end{document}. Nonetheless, we prove a weaker estimate for such dimensions as well.

Strictly positive definite non-isotropic kernels on two-point homogeneous manifolds: the asymptotic approach

We present sufficient conditions for a family of positive definite kernels on a compact two-point homogeneous space to be strictly positive definite based on their expansion in eigenfunctions of the Laplace–Beltrami operator. We also present a characterisation of this kernel class. The family analyzed is a generalization of the isotropic kernels and the case of a real sphere is analyzed in details.

Point pattern analysis and classification on compact two-point homogeneous spaces evolving time.

This paper introduces a new modeling framework for the statistical analysis of point patterns on a manifold defined by a connected and compact two-point homogeneous space, including the special case of the sphere. The presented approach is based on temporal Cox processes driven by a -valued log-intensity. Different aggregation schemes on the manifold of the spatiotemporal point-referenced data are implemented in terms of the time-varying discrete Jacobi polynomial transform of the log-risk process. The n-dimensional microscale point pattern evolution in time at different manifold spatial scales is then characterized from such a transform. The simulation study undertaken illustrates the construction of spherical point process models displaying aggregation at low Legendre polynomial transform frequencies (large scale), while regularity is observed at high frequencies (small scale). K-function analysis supports these results under temporal short, intermediate and long range dependence of the log-risk process.

Strong Local Nondeterminism and Exact Modulus of Continuity for Isotropic Gaussian Random Fields on Compact Two-Point Homogeneous Spaces

This paper is concerned with sample path properties of real-valued isotropic Gaussian fields on compact two-point homogeneous spaces. In particular, we establish the property of strong local nondeterminism of an isotropic Gaussian field and then exploit this result to establish an exact uniform modulus of continuity for its sample paths.

Series expansions among weighted Lebesgue function spaces and applications to positive definite functions on compact two-point homogeneous spaces

Geodesically isotropic positive definite functions on compact two-point homogeneous spaces of dimension d have series representation as members of weighted Lebesgue spaces L1w([−1,1]), where the weight w(x)=wα,β(x)=(1−x)α(1+x)β is the one related to the Jacobi orthogonal polynomials P(α,β)(x) in [−1,1], and the exponents α and β are related to the dimension d. We derive some recurrence relations among the coefficients of the series representations under different exponents, and we apply them to prove inheritance of positive definiteness between dimensions. Additionally, we give bounds on the curvature at the origin of such positive definite functions with compact support, extending the existing solutions from d-dimensional spheres to compact two-point homogeneous spaces.

Stochastic comparison for elliptically contoured random fields

This paper presents necessary and sufficient conditions for the peakedness comparison and convex ordering between two elliptically contoured random fields about their centers. A somewhat surprising finding is that the peakedness comparison for the infinite dimensional case differs from the finite dimensional case. For example, a Student’s t distribution is known to be more heavy-tailed than a normal distribution, but a Student’s t random field and a Gaussian random field are not comparable in terms of the peakedness. In particular, the peakedness comparison and convex ordering are made for isotropic elliptically contoured random fields on compact two-point homogeneous spaces.

Concentration estimates for finite expansions of spherical harmonics on two-point homogeneous spaces via the large sieve principle

We study the concentration problem on compact two-point homogeneous spaces for finite expansions of eigenfunctions of the Laplace–Beltrami operator using large sieve methods. We derive upper bounds for concentration in terms of the maximum Nyquist density. Our proof uses estimates of the spherical harmonics basis coefficients of certain zonal filters and an ordering result for Jacobi polynomials for arguments close to one.

Regularity and approximation of Gaussian random fields evolving temporally over compact two-point homogeneous spaces

We consider Gaussian random fields on the product of a compact two-point homogeneous space cross the time, which are space isotropic and time stationary. We study regularity properties of these random fields in terms of function spaces whose elements have different smoothness in the space and time domain. Namely, we express the norm of the corresponding covariance kernel functions in terms of the summability of the associated spectral coefficients. Furthermore, we define an approximation method based on the truncation of the expansion related to the spectral representation of a given random field. The accuracy of this approximation is measured in the $$L^p$$ sense. Finally, we model a space–time dataset of ozone concentrations in Mexico City using a seasonal temporal covariance function constructed through an expansion of Jacobi polynomials. We find that we need relatively few Jacobi polynomials to get the best fit to the data in terms of the deviance information criterion. We discuss the characteristics of this model, including seasonality, decay and approximate conditional independencies.

Regularity, continuity and approximation of isotropic Gaussian random fields on compact two-point homogeneous spaces

Gaussian random fields defined over compact two-point homogeneous spaces are considered and Sobolev regularity and Hölder continuity are explored through spectral representations. It is shown how spectral properties of the covariance function associated to a given Gaussian random field are crucial to determine such regularities and geometric properties. Furthermore, fast approximations of random fields on compact two-point homogeneous spaces are derived by truncation of the series expansion, and a suitable bound for the error involved in such an approximation is provided.

Positive Definiteness on Products of Compact Two-point Homogeneous Spaces and Locally Compact Abelian Groups

AbstractIn this paper, we consider the problem of characterizing positive definite functions on compact two-point homogeneous spaces cross locally compact abelian groups. For a locally compact abelian group $G$ with dual group $\widehat{G}$, a compact two-point homogeneous space $\mathbb{H}$ with normalized geodesic distance $\unicode[STIX]{x1D6FF}$ and a profile function $\unicode[STIX]{x1D719}:[-1,1]\times G\rightarrow \mathbb{C}$ satisfying certain continuity and integrability assumptions, we show that the positive definiteness of the kernel $((x,u),(y,v))\in (\mathbb{H}\times G)^{2}\mapsto \unicode[STIX]{x1D719}(\cos \unicode[STIX]{x1D6FF}(x,y),uv^{-1})$ is equivalent to the positive definiteness of the Fourier transformed kernels $(x,y)\in \mathbb{H}^{2}\mapsto \widehat{\unicode[STIX]{x1D719}}_{\cos \unicode[STIX]{x1D6FF}(x,y)}(\unicode[STIX]{x1D6FE})$, $\unicode[STIX]{x1D6FE}\in \widehat{G}$, where $\unicode[STIX]{x1D719}_{t}(u)=\unicode[STIX]{x1D719}(t,u)$, $u\in G$. We also provide some results on the strict positive definiteness of the kernel.

Approximation tools and decay rates for eigenvalues of integral operators on a general setting

We provide the characterization of the Peetre-type K-functional on a compact two-point homogeneous space in terms of the rate of approximation of a family of multipliers operators. This extends the well known results on the spherical setting. The characterization is employed to prove that an abstract Holder condition or finite order of differentiability assumption on generating positive kernels of integral operators implies a sharp decay rates for their eigenvalues sequences. Consequently, sharp upper bounds for the Kolmogorov n-width of unit balls in reproducing kernel Hilbert space are obtained.

Isotropic Covariance Matrix Functions on Compact Two-Point Homogeneous Spaces

The covariance matrix function is characterized in this paper for a Gaussian or elliptically contoured vector random field that is stationary, isotropic, and mean square continuous on the compact two-point homogeneous space. Necessary and sufficient conditions are derived for a symmetric and continuous matrix function to be an isotropic covariance matrix function on all compact two-point homogeneous spaces. It is also shown that, for a symmetric and continuous matrix function with compact support, if it makes an isotropic covariance matrix function in the Euclidean space, then it makes an isotropic covariance matrix function on the sphere or the real projective space.

Time-Varying Isotropic Vector Random Fields on Compact Two-Point Homogeneous Spaces

A general form of the covariance matrix function is derived in this paper for a vector random field that is isotropic and mean square continuous on a compact connected two-point homogeneous space and stationary on a temporal domain. A series representation is presented for such a vector random field which involves Jacobi polynomials and the distance defined on the compact two-point homogeneous space.

Strictly Positive Definite Functions on Compact Two-Point Homogeneous Spaces: the Product Alternative

For two continuous and isotropic positive definite kernels on the same compact two-point homogeneous space, we determine necessary and sufficient conditions in order that their product be strictly positive definite. We also provide a similar characterization for kernels on the space-time setting $G \times S^d$, where $G$ is a locally compact group and $S^d$ is the unit sphere in $\mathbb{R}^{d+1}$, keeping isotropy of the kernels with respect to the $S^d$ component. Among other things, these results provide new procedures for the construction of valid models for interpolation and approximation on compact two-point homogeneous spaces.

Upper bounds for s-distance sets and equiangular lines

The set of points in a metric space is called an s-distance set if pairwise distances between these points admit only s distinct values. Two-distance spherical sets with the set of scalar products {α,−α}, α∈[0,1), are called equiangular. The problem of determining the maximum size of s-distance sets in various spaces has a long history in mathematics. We suggest a new method of bounding the size of an s-distance set in compact two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in Rn, n≥7, is n(n+1)2 with possible exceptions for some n=(2k+1)2−3, k∈N. We also prove the universal upper bound ∼23na2 for equiangular sets with α=1a and, employing this bound, prove a new upper bound on the size of equiangular sets in all dimensions. Finally, we classify all equiangular sets reaching this new bound.

A limit formula for semigroups defined by Fourier-Jacobi series

I. J. Schoenberg showed the following result in his celebrated paper [Schoenberg, I. J., Positive definite functions on spheres. Duke Math. J. 9 (1942), 96-108]: let ⋅ \cdot and S d S^d denote the usual inner product and the unit sphere in R d + 1 \mathbb {R}^{d+1} , respectively. If F d \mathcal {F}^d stands for the class of real continuous functions f f with domain [ − 1 , 1 ] [-1,1] defining positive definite kernels ( x , y ) ∈ S d × S d → f ( x ⋅ y ) (x,y)\in S^d \times S^d \to f(x\cdot y) , then the class ⋂ d ≥ 1 F d \bigcap _{d\geq 1} \mathcal {F}^d coincides with the class of probability generating functions on [ − 1 , 1 ] [-1,1] . In this paper, we present an extension of this result to classes of continuous functions defined by Fourier-Jacobi expansions with nonnegative coefficients. In particular, we establish a version of the above result in the case in which the spheres S d S^d are replaced with compact two-point homogeneous spaces.

Covering numbers of isotropic reproducing kernels on compact two‐point homogeneous spaces

AbstractIn this paper we present upper and lower estimates for the covering numbers of the unit ball of a reproducing kernel Hilbert space associated to a continuous isotropic kernel on a compact two‐point homogeneous space (CTPHS). These estimates are obtained from estimates on the decay of the Fourier–Jacobi coefficients of the kernel via applications of the Funk–Hecke formula and the Schoenberg series representation of an isotropic kernel on CTPHS and also by the use of cubature formulas on these spaces.

Strict positive definiteness on products of compact two-point homogeneous spaces

ABSTRACTWe present an explicit characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of compact two-point homogeneous spaces, covering almost all possible choices for the spaces. The result complements similar characterizations previously obtained for products of high-dimensional spheres.

Strict positive definiteness of multivariate covariance functions on compact two-point homogeneous spaces

The authors provide a characterization of the continuous and isotropic multivariate covariance functions associated to a Gaussian random field with index set varying over a compact two-point homogeneous space. Sufficient conditions for the strict positive definiteness based on this characterization are presented. Under the assumption that the space is not a sphere, a necessary and sufficient condition is given for the continuous and isotropic multivariate covariance function to be strictly positive definite. Under the same assumption, an alternative necessary and sufficient condition is also provided for the strict positive definiteness of a continuous and isotropic bivariate covariance function based on the main diagonal entries in the matrix representation for the covariance function.

Strictly positive definite kernels on compact two-point homogeneous spaces

Strictly positive definite kernels on compact two-point homogeneous spaces